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This Would Have Gotten Me to Pay Better Attention In Trigonometry

Via applied mathemagics.

Technorati Tags: tan,line,tanget,curve

“Reverse Phone Book” for Algebra

I came across an interesting site called RIES. If you give it a number, it returns equations that approximate that number.

Weird, eh? Usually you have a tough equation, and you approximate a solution to it with a number.

For example, I input 42, and RIES outputs these equations that can be solved for x and which yield an answer close to 42:

• x+7=7^2, an exact solution
• 3x=5^3, approximation is accurate within 1%
• (x+5)^2=3*7, , approximation is accurate within 1%
• x^5=8^9, approximation is accurate within 1%
• 5(x+1)=6^3, approximation is accurate within 1%

An odd thing for someone who isn’t a mathematician: this is sort of like a reverse phone book for algebra.

Via I Love Charts via the details in the fabric via XKCD.

Technorati Tags: reverse,phone,book,algebra,ries

A Joke for Coders

Test your code before it matters. This goes for your spreadsheets too … if you didn’t’ think they count as “code”.

Technorati Tags: excel,test,code,production

Political Correctness and the Rate of Return from Schooling

Wow. Bryan Caplan points out that economists who measure the returns from schooling are aware that individual ability is an important determinant, but suppress correcting for its effects because it is measured badly.

This is like not using batting average (at all) to compare baseball players because there is much more to the game.

Repeating the paraphrase of Orwell “Some ideas are so stupid that only intellectuals believe them.”*

Here’s Bryan’s result:

… If you mention ability bias, however, labor economists will quickly point you to a massive literature that supposedly debunks it.

But if you pay close attention, there's a bizarre omission. Despite their mighty debunking efforts, labor economists almost never test for ability bias in the most obvious way: Measure ability, then re-estimate the return to education after controlling for measured ability. For example, you could measure IQ, then estimate the return to education after controlling for IQ.

When I ask labor economists about their omission, they have a puzzling response: "IQ is a very incomplete measure of ability." True enough. But the right lesson to draw is that controlling for IQ provides a lower bound for the severity of ability bias. After all, if the estimated return to education falls sharply after controlling for just one measure of ability, imagine how much it might fall after controlling for measures of all ability.

What happens to the return to education after controlling for IQ? I've done the statistics myself on the NLSY, and found that the estimated return to education falls by about 40% …

So, the reported returns to education — how much extra one gets paid for more schooling — are almost double the corrected estimates because they’re ignoring the basic observation that smart people are likely to get paid more anyway.

The flip side of this is that the value of teachers and schools is overstated by ignoring the quality of the students put into the system.

Gee … do’ya think there’s any grant funding out there interested in establishing and promoting that result?

Statistically, Bryan is pointing out that conventional parameter estimates are biased upward, but he’s missing a second problem. The big three problems in statistics are bias, consistency, and efficiency — and the conventional estimates are not just biased, they’re also inefficient.

Inefficiency is a lot more subtle than bias. Frankly, I don’t think I understood it until about 8 years into my academic career when I had to explain in seminars to finance Ph.D.’s why they should listen to an economics Ph.D. about the problem with one of their techniques.

In short, an inefficient estimator is one that’s just dumb: like forecasting the weather without looking at the sky. There’s a lot more to it than that, but it means that you’re not doing something which might be helpful.

One way to think about this is that bias is about parameter estimates that are off in one direction or the other, while efficiency is about standard error estimates that are off in one direction or the other.

In practice, this suggests that not only are the typical estimates of the returns to education biased upwards, but they probably reported to be a lot more precise than they actually are. That is, Bryan is pointing out that the parameter estimates are biased upward, and I’m pointing out that the standard error estimates are probably biased downwards.

Bias and inefficiency also have distributional consequences for the conventional results. Because those results are based on leaving something out, they suggest that the things that are left in are more important than they would be in a better model. The thing is, you don’t know which of your variables is more or less seriously affected until you run the better model. Maybe you’re lucky, and the variable of interest is the one that’s least affected … and maybe you’re not.

* Neo-neocon points out that the actual quote from Orwell’s Notes On Nationalism is “One has to belong to the intelligentsia to believe things like that: no ordinary man could be such a fool”.

Technorati Tags: returns,schooling,education,ability,bias,inefficiency

Confidence Interval

Technorati Tags: confidence,interval

30 Ballparks In 5 Weeks

Start today in Kansas City … and drive over 18K miles over the next 35 days and you can see a baseball game in every major league ballpark. With no doubleheaders!

How does one figure this out? Linear programming — the old standard of management science.

This is an example of a network programming problem, as in Chapter 12 of the Powell and Baker MBA text.

The decisions form an array: 30 ballparks against some number of days (larger than the likely minimum number).

The objective is probably to minimize the number of days (although it could be driving miles too).

The constraints are that 1) each park is visited one time, 2) there is a game at the park that day, 3) on each day there is at most one game to be seen, 4) drive time between games must be possible in 12 hour stints with 8 hours off in between (there’s no mention of the highway speed assumed).

The article from the June 3 issue of The Wall Street Journal says:

… Blatt plugged the schedule for each team into his model, while setting one key parameter: For every 12 hours of estimated driving between ballparks, the system must allow for eight hours of rest, ensuring that the road trip is, at least in theory, humanly possible.

If you think about it, there are probably a lot of ways to get to 30 parks in 30 days. But most of those would be pointless ones, involving trips from NYC to LA to Philly to Seattle on consecutive days, and so on.

That “key parameter” is what makes the trip “humanly possible” by car, because it gives you 45 hours to do the longest trip (1,530 miles) from Houston to LA, 39 hours to do the 1,330 from Seattle to Colorado, and 27 hours to do the 1,047 miles from Colorado to Milwaukee — and you only get one break on that last one. After that, the rest look ease.

Technorati Tags: linear,programming,baseball,thirty,ballparks,road,trip